Pump-probe spectroscopy of chiral vibrational dynamics

A planar molecule may become chiral upon excitation of an out-of-plane vibration, changing its handedness during half a vibrational period. When exciting such a vibration in an ensemble of randomly oriented molecules with an infrared laser, half of the molecules will undergo the vibration phase-shifted by π compared to the other half, and no net chiral signal is observed. This symmetry can be broken by exciting the vibrational motion with a Raman transition in the presence of a static electric field. Subsequent ionization of the vibrating molecules by an extreme ultraviolet pulse probes the time-dependent net handedness via the photoelectron circular dichroism. Our proposal for pump-probe spectroscopy of molecular chirality, based on quantum-chemical theory and discussed for the example of the carbonyl chlorofluoride molecule, is feasible with current experimental technology.


A. Geometry of the envisioned experiment
The proposed experiment is sketched in Figure S1: Two oppositely charged electrodes (indicated by the red and green rings with "+" and "−" for the charges) create a static electric field with strength E static . A circularly polarized UV pulse, for example the 4th harmonic of a Ti:Sa laser, is sent to a sample of planar molecules such that its polarization, specified by the two orthogonal electric fields E L1 and E L2 , is orthogonal to the static electric field. Hence the triple product E static · [E L1 × E L2 ] ̸ = 0. The electric field tends to orient the molecules by their permanent dipole moment µ 00 , while the UV pump pulse creates a coherent chiral vibrational wavepacket by resonant Raman excitation of the out-of-plane vibration. The resulting time-dependent chirality can be probed by a circularly polarized XUV probe pulse with the same propagation direction as the UV pump pulse. The probe ionizes the molecular ensemble, and photoelectron angular distributions (PAD) are measured using velocity map imaging (VMI). Recording PADs with both helicities of the probe pulse and taking their difference allows for quantifying the produced enantiomeric excess in terms of photoelectron circular dichroism (PECD). Figure S1 shows one possible configuration that would allow to use the static electric field of the VMI for E static . Other field configurations might exist that are more suitable for a given experimental setup.

+ -
Pump pulse Probe pulse Electrons Figure S1 Example of a field configuration to realize the proposed pump-probe spectroscopy in an experiment. Electric field coils create the static electric field as well as the extraction field to image the photoelectrons resulting from the ionization of the planar molecules, taken to be COFCl here. The UV pump pulse prepares the vibrational superposition via a Raman excitation, whereas the probe pulse ionizes the (time-dependent) superposition.

B. Minimum number of fields to realize enantiomeric excess
To produce a broken parity, chiral state in a molecule, we need to create a superposition state from two (or more) levels of opposite parity [20]. One example of such states can be the vibrational states of the molecule, for instance the ground (|0⟩) and first excited (|1⟩) states of the out-of-plane (OOP) vibration of the COFCl molecule (for definition of the OOP coordinate see also Fig. S3). Let us assume the molecule to be initially in state |0⟩. The superposition state after the excitation, for a single molecule with a particular orientation in space, is given by where ℏω 01 the energy difference between the two levels. The coefficients c (n) 1 are calculated with time-dependent perturbation theory of order n, c (n) Here, ℏω v1 is the energy difference between the vibrational states |v ̸ = 1⟩ and |1⟩. In particular, we consider direct excitation of the vibrational state |1⟩ with an IR-pulse, which corresponds to first order perturbation theory (n = 1), and, alternatively, UV Raman excitation via virtual excited states, which is described by second order perturbation theory (n = 2). The laser-molecule interaction is given bŷ whereμ is the dipole moment in the molecular frame, E(t) the electric field in the laboratory frame, and R the rotation matrix connecting the two frames and depending on the Euler angles ϕ, θ, χ, For excitation with an IR pulse (n = 1) or a Raman pulse (n = 2), inserting Eq. (3) into Eq. (2) allows us to separate c wherec (n) 1 (t) does not depend on the Euler angles and is calculated in detail in Sec. IC. The angular dependence is expressed by the product n k=1 E k ·R (φ, θ, χ) µ k , where E k denotes the polarization direction and amplitude of the field responsible for the kth order interaction, and µ k = ⟨v ′ |μ|v⟩ is the corresponding transition matrix element. The time-dependent partc (n) 1 (t) does not depend on the Euler angles and is calculated in detail in Sec. I.C. For simplicity, we assume here that in case of the UV Raman excitation the coefficients c  Observables for a molecule with given orientation are obtained as We are interested in observables that are sensitive to the coherence of the state and assume without loss of generality ⟨0|Ô|0⟩ = ⟨1|Ô|1⟩ = 0 and ⟨0|Ô|1⟩ ̸ = 0. Then with c (n) 1 = |c (n) 1 | · exp(−iφ 1 ). In order to relate ⟨O⟩(t), i.e., the signal produced by a single molecule at a given orientation, to the macroscopic response of the system, we need to average over all the orientations that a molecule in the gaseous sample can take. We describe the rotational average classically, where dΩ = 2π 0 dϕ π 0 sin θdθ 2π 0 dχ denotes the integration over the Euler angles. In the presence of a static electric field E static , the orientational distribution of the molecules in thermal equilibrium at temperature T is given by where Q is the partition function, E rot the (classical) rotational energy and µ 00 the permanent dipole moment of the molecule. Classical rotational averaging is valid as long as as the rotational spectrum is not resolved, i.e., for time-scales shorter than the rotational period. The rotational period of the molecules around the principle axis α (α = a, b, c) can be estimated as τ rot,α ∝ B −1 α , where B α is the rotational constant for rotation about that principle axis. The shortest rotational period corresponds to the largest rotational constant, B a = A. For COFCl, A = 12 GHz, and thus τ rot,a = 8 ps. This is much longer than typical pump-probe delays of a few hundred femtoseconds. Moreover, we neglect a possible time-dependence of P (ϕ, θ, χ). This is justified if the interaction with the pump pulse is too weak to create a rotational wavepacket that causes pronounced alignment. To test this assumption, we have calculated the classical alignment factors of COFCl with a Monte-Carlo simulation for classical rigid rotors. We have found that a pump pulse with an intensity of 10 13 W/cm 2 and a FWHM of 150 fs indeed does not create molecular alignment within several ps, so that the assumption of a time-independent angular distribution function is justified.
Expanding the Boltzmann distribution to first order in the static field amplitude, the orientational distribution can be approximated by where is the field-free isotropic distribution which does not depend on the Euler angles, while describes the first-order orientation produced by the static field. Rotational Debye temperatures for COFCl, below which the thermal ensemble cannot be described classically anymore, are around 0.5 K. For rotational temperatures above 1 K, a classical treatment is thus well justified. As a result, the orientational average for an observable O is given by with E n+1 = E static and µ n+1 = µ 00 . Evaluating the integrals according to Ref. [22], we find, depending on the number of external fields, N =1: dΩ(E 1 · R(ϕ, θ, χ)µ 1 ) = 0. This means that a single IR pulse (with E 1 = E L1 and µ 1 = µ 01 in the notation of the main text), inducing the transition |0⟩ → |1⟩, cannot produce a chiral signature for the ensemble, . Such a configuration could be realized in two different ways with both resulting, however, in a vanishing integral: IR excitation |0⟩ → |1⟩ combined with static field orientation (E 1 == E L1 , E 2 = E static , µ 1 = µ 01 , and µ 2 = µ 00 in the notation of the main text). The integral vanishes because the permanent dipole moment µ 00 is in the molecular plane, whereas the transition dipole moment µ 01 is orthogonal to the molecular plane and therefore (µ 00 · µ 01 ) = 0. Raman excitation |0⟩ → |1⟩ without static field also does not yield a chiral signal (E 1 = E L1 , E 2 = E L2 , µ 1 = µ 0v , and µ 2 = µ v1 in the notation of the main text). The integrand vanishes because the two transition dipole moments are necessarily orthogonal to each other, as explained in more detail below in Section I D.
. These triple products do not necessarily vanish, which is why N = 3 is the smallest number of fields that may result in a non-zero rotational average. It requires three orthogonal (permanent or transition) dipole moments, with their triple product changing sign under spatial inversion. One possibility to realize the corresponding field configuration is Raman excitation with circular polarization in the presence of a static electric field perpendicular to the polarization plane of the circularly polarized field. E 1 = E L1 , E 2 = E L2 , E 3 = E static , µ 1 = µ 0v , µ 2 = µ v1 , and µ 3 = µ 00 in the notation of the main text.

C. Magnitude of the induced enantiomeric excess
After clarifying the conditions for observing enantiomeric excess in a sample of randomly oriented molecules, we now estimate its magnitude. To this end, we utilize time-dependent perturbation theory (TDPT). The electric field driving the Raman exciation is given by with E L1 = E L · n 1 and E L2 = E L · n 2 . Here, E L is the electric field peak amplitude, n and φ are the polarization axes and phases, ω L is the pulse carrier frequency, and τ the pulse duration, related to the full width at half maximum (FWHM) by FWHM = 2 ln(2)τ . We consider Raman excitation from the ground to the first excited state of the out-of-plane vibration in the electronic ground state via vibronically excited states |v⟩, |0⟩ → |v⟩ → |1⟩, see Fig. S2. This requires TDPT to second order, with the corresponding wavefunction given by where ω lm = (E m − E l )/ℏ is the angular frequency of the |l⟩ → |m⟩ transition. Invoking the rotating wave approximation, the expansion coefficients become where ∆ lm = ω lm − sgn(ω lm ) · ω L is the detuning and µ lm the transition dipole moment in the molecular frame, and R = R(ϕ, θ, χ) is the rotation matrix (Eq. (4)), transforming the molecular frame into the laboratory fame. Evaluation to first order yields the coefficients of the virtual states |v⟩, represents the spectral profile of the Gaussian-shaped laser pulse. The second order coefficient cannot be evaluated analytically for non-vanishing detuning. We therefore estimate it approximately for zero detunings and account for the spectral pulse profile by multiplication with the two exponents of the form exp − τ 2 ∆ 2 4 afterwards. This procedure is equivalent to treating the excitation as resonant absorption at the specific transition frequencies within the broad Gaussian spectrum of the pulse. With the time integral of the form this results in Inserting Eq. (15) into Eq. (12) yields rotationally averaged expectation values in second order TDPT, Equation (16) reveals how to maximize enantiomeric excess: The Raman pulse should be circularly polarized since | sin(φ 2 − φ 1 )| = 1 is maximal for |φ 2 − φ 1 | = π/2, and the static electric field and polarization axes should all be mutually orthogonal to maximize n 0 · [n 1 × n 2 ] = 1.

D. Symmetry considerations
In this Section, we employ symmetry arguments to determine which components of the dipole and transition dipole moments µ 00 , µ 0v and µ v1 are non-zero. We consider planar molecules of C s symmetry, with the ab-plane as symmetry plane and the molecular c-axis perpendicular to the symmetry-plane, then, the molecular axes a, b ∈ A ′ and c ∈ A ′′ .
The Cartesian components of the molecular dipole moments are the projections onto the molecular axes, µ a ∝ a, µ b ∝ b, µ c ∝ c, and thus transform as the molecuar axis a, b and c. In order to determine the non-zero components of the dipole and transition dipole moments, we separate the OOP vibronic states |0⟩, |1⟩, and |v⟩ into their vibrational and electronic part, The out-of-plane motion has A ′′ symmetry, which means that the symmetry of the vibrational states along the OOP coordinate is Γ(|0⟩ vib ) = A ′ and Γ(|1⟩ vib ) = A ′′ , i.e. the ground vibrational state is symmetric while the first excited state is anti-symmetric. Likewise, for the vibrational wavefunctions in the excited electronic state Γ(|v⟩ vib ) is either A ′ or A ′′ for even v and odd v, respectively.
The components i, j, k = a, b, c of the dipole and transition dipole moments are then For each of the components to be nonzero, the product of the respective representations has to be fully symmetric, i.e. transform according to The excited electronic state is antisymmetric (Γ(|e⟩ el ) = A ′′ ) because the electronic transition is n → π * , and the π-orbitals are antisymmetric with respect to the plane of the molecule. The symmetry of dipole moment components is the same as for the molecular frame axes, i.e., µ a , µ b ∈ A ′ , whilst µ c ∈ A ′′ . Thus, for the permanent dipole we have only µ a 00 and µ b 00 available for being nonzero, and µ c 00 = 0. If the virtual state is symmetric Γ(|v⟩ vib ) = A ′ , then the requirement is Γ(µ j 0v ) = Γ(μ j ) ⊗ A ′′ and Γ(µ k v1 ) = Γ(μ k ) ⊗ A ′ . These conditions can be fulfilled if j = c and k = a, b. Similarly, if Γ(|v⟩ vib ) = A ′′ , then the conditions are inverted Γ(µ j 0v ) = Γ(μ j ) ⊗ A ′ and Γ(µ k v1 ) = Γ(μ k ) ⊗ A ′′ , which is fullfiled by j = a, b and k = c.
Orientation-averaged enantiomeric excess requires the triple product to be nonzero where we have used Einstein notation with ε ijk the Levi-Civita symbol, and i, j, k = a, b, c. Since ε ijk ̸ = 0 only for i ̸ = j ̸ = k ̸ = i, the product is given by three different dipole moment components and these must be allowed to be non-zero by symmetry. Therefore, the following combinations are allowed by symmetry: can be nonzero.

II. COMPUTATIONAL DETAILS FOR THE COFCL MOLECULE
A. Electronic structure The quantum-chemical computations were performed at the B3LYP/cc-pVTZ level of theory [38,39,40,42] using the Orca 4 package [44]. The structure of COFCl was optimized in the ground electronic state, yielding the equilibrium structure of the molecule given in Table S1 below. The harmonic frequency calculations confirmed the minimum. The ten lowest excited states were computed using time-dependent density functional theory within the Tamm-Dancoff approximation [43]. The molecular frame was chosen such that all atoms lie in the ab plane. The out-of-plane (OOP) coordinate ξ corresponds the c-coordinate of the carbon atom, while all the other atoms (O, F, and Cl) are assumed to be frozen at their equilibrium positions, cf. Figure S3. For the ground, first excited, and ground ionic potential energy surfaces (PES), 100 points were computed with ξ ∈ [−0.9Å, +0.9Å]. Along with the energies, the ground state dipole moment µ g (ξ) = ⟨g(ξ)| elμ |g(ξ)⟩ el and transition dipole moment µ ge (ξ) = ⟨g(ξ)| elμ |e(ξ)⟩ el were computed at each ξ. To provide the quantum-chemical results here in a compact form, the energies V (ξ) and dipole moments µ(ξ) were approximated by polynomials The approximation results are presented below in Section IV, with the polynomial coefficients listed in Tables S2, S3, and S4. The quality of the approximation can be judged from Figures S9, S10, and S11.

B. Vibrational structure
To calculate the vibrational states |n⟩ vib of the OOP motion in the ground and first excited state, the grid points used in the electronic structure calculations were symmetrized along the mirror plane (ξ = 0) and then interpolated to a uniform grid of 1000 points using cubic splines. The reduced mass for the OOP motion is given by With these parameters, the vibrational Schrödinger was solved for both electronic manifolds, using a discrete-variable representation [45]. The vibrational states in the electronically excited state |v⟩ vib were the symmetrized with respect to ξ = 0. From the resulting vibronic wavefunctions, the vibrationally averaged permanent dipole moments µ 00 = ⟨0| vib µ g (ξ)|0⟩ vib = ⟨0| vib ⟨g| el |μ|g⟩ el |0⟩ vib and transition dipole moments µ 0v = ⟨0| vib µ ge (ξ)|v⟩ vib = ⟨0| vib ⟨g| el |μ|e⟩ el |v⟩ vib , µ 1v = ⟨1| vib µ ge (ξ)|v⟩ vib = ⟨1| vib ⟨g| el |μ|e⟩ el |v⟩ vib , corresponding to the Raman transition shown in Figure S2, were computed. The quality of the vibronic states was evaluated by comparison with the available experimental data for COF 35 Cl. The experimental value for the out-of-plane fundamental vibrational transition (|0⟩ → |1⟩) is 667 cm −1 , the value obtained in our calculations was 662 cm −1 (relative error of 1%). From the |µ 0v | 2 values, the UV absorption spectrum was also calculated. The comparison with the available band shape from the MPI-Mainz UV/VIS Spectral Atlas [50] shows a good reproduction of the first UV band, cf Fig. S4.

C. Vibrationally averaged photoelectron angular distributions
For the calculation of photoelectron angular distributions (PAD), we have extended the procedure of Refs. [29,30] to explicitly account for the vibrational dynamics via vibrational averaging. Starting in the Born-Oppenheimer approximation, we write the vibronic state of the molecule as where |g⟩ el refers to the electronic ground state, |e⟩ el to the electronically excited state, and |k⟩ el is the electronic state of the ion after photoionization with k = (k, ϕ k , θ k ) the laboratory frame momentum of the emitted electron. |v⟩ vib represent the vibrational eigenstates in the corresponding PES with eigenfrequencies ω v,n . The expansion coefficients of the electronic ground state, c j,g (t) = n c (n) j,g (t) (with c (n) j,g (t) introduced in Sec. I C), describe the chiral vibrational wavepacket. We evaluate them up to second-order time-dependent perturbation theory according to Equation (14), describing the Raman excitation by the pump pulse. Here, we solve the second-order time integral numerically. Assuming one-photon ionization from the highest occupied molecular orbital by the circularly polarized XUV probe pulse E I , we calculate the ionic expansion coefficients in Equation (18) by third-order perturbation theory, where R(ϕ, θ, χ) transforms the molecule-fixed dipole operator into the laboratory-fixed frame of reference. Hence, the expansion coefficients depend on the orientation of the molecule, i.e., c v,n (t) ≡ c v,n (t; ϕ, θ, χ) for n = g, e, k. Equation (19) implies an integral over the vibrational coordinate, where χ j,g (ξ) and χ v ′ ,k (ξ) are the vibrational eigenfunctions in the neutral and ionic ground electronic states. In other words, the vibrational dynamics is taken into account by averaging over the parametric dependence of the electronic states on the nuclear coordinates, with the approximation that all nuclei are fixed except for the OOP motion, see Fig. S3. For a given, fixed value of the OOP coordinate ξ, we follow the procedure of Refs. [29,30] to evaluate the electronic dipole matrix elements in Equation (19), µ kg (ξ) = ⟨k(ξ)| elμ |g(ξ)⟩ el , by applying a single center expansion. This yields the transition dipole moments to the electronic continuum state of the ion in the frozen-core Hartree-Fock approximation. The calculation is performed with the ePolyScat program. It requires the electronic ground state to be represented by a single Slater determinant [54,55], which we calculate at the HF/aug-cc-pVTZ level of theory. Thus, the continuum dipole matrix elements are calculated for different values of ξ by manipulating the molecular geometry in the calculation of the ground state Slater determinant. This enables us to obtain vibronic transition dipole matrix elements in terms of the overlap of vibrational nuclear wavefunctions, calculated as described in Sec. II B, and the parametrized electronic dipole matrix elements. We use partial waves up to L = 80 to represent the bound orbitals on the angular grid in order to ensure converged transition dipole matrix elements at photoelectron energies up to 20 eV. In order to reduce the numerical effort in evaluating Equation (19), we approximate the shape of the ionizing pulse by δ(t − t I ) and assume the PAD to display a Gaussian energy dependence, which is given by the ionizing pulse spectrum [56]. This neglects the energy dependence of the continuum dipole matrix elements. We have verified that this is a reasonable approximation as long as the probe pulse duration is significantly shorter than the oscillation period of enantiomeric excess, i.e., the ionization pulse is not longer than 5 fs. Of course, the latter is required in any case, in order to resolve the chiral vibrational motion.
For a single molecule with given orientation, the laboratory-frame PAD at time t for a photoelectron with momentum k is given in terms of the ionic expansion coefficients, where the sum runs over all populated vibrational states v ′ in the ground electronic state of the ion. The orientationaveraged laboratory-frame PAD is then obtained by integrating over the Euler angles, where P (ϕ, θ, χ) describes the orientational distribution of the molecules, including the effect of the static electric field, cf. Sec. I B. As explained in Sec. I B, it is well justified to take the orientational distribution to be time-independentthis applies also to duration of the probe pulse. The integral over molecular orientations in Equation (21) is evaluated numerically. Photoelectron circular dichroism (PECD) is the forward-backward asymmetry visible in the dichroic difference of PADs upon ionization with left-and right-circularly polarized (LCP/RCP) light. We quantify PECD according to the convention of Ref. [24], i.e.
This convention is commonly used in experiments and often combined with an integration of the PADs over the forward and backward hemisphere [12,25,26,27]. The latter can result in smaller values, if the individual hemispheres contain contributions to PECD with opposite signs, which cancel each other upon integration. In the proposed experiment, we observe almost complete cancellation of PECD inside the hemispheres with the pulse parameters given in the main text. In our understanding, this cancellation happens by chance for COFCl at the selected photoelectron energy, since there is no symmetry requirement that forces this cancellation to occur. The orientation-averaged PADs are cylindrically symmetric due to the cylindrical symmetry of the proposed experimental setup (cf. Fig. S1). Hence, it does not matter if PECD is calculated from the PADs itself or from velocity map images (VMI), i.e. the projection of Eq. (22) onto the yz-plane. However, due to the orientation of the molecules by the static electric field, the dichroic differences are not antisymmetric with respect to the light propagation direction as it is the case for randomly oriented molecules. For completeness, Fig. S5 shows polar plots of the orientationaveraged laboratory frame PADs for the pump-probe delays that are marked in Fig. 3 Figure S5 Photoelectron angular distribution: Polar plots of the orientation-averaged laboratory frame photoelectron angular distributions obtained with a left-and right-circularly polarized ionizing pulse and the corresponding dichroic difference where θ k is the polar angle of the photoelectron momentum. The latter has been normalized with respect to the mean value of the corresponding PADs according to Eq. (22). The simulation parameters are the same as for the results presented in Fig. 3 of the main text. The rows correspond to the time delays that are marked with dashed vertical lines in Fig. 3 of the main text. The dichroic difference in the right column is shown as absolute value in percent, while the sign is encoded in the color, i.e., red and blue corresponds to positive and negative values, respectively.

III. DEPHASING AND RELAXATION TIMESCALES
An essential aspect of the ultrafast experiments is the decay time of the signal, because it limits the timescale over which the probe can monitor the effect induced by the pump pulse. Here, we consider mechanisms that may limit observation of the vibrational coherence. To observe the motion of the chiral wavepacket, at least one period of the out-of-plane vibration in COFCl, which is about 50 fs. Ideally, the shortest dephasing time should be at least an order of magnitude larger. Since we also assume pump and probe not to overlap in time, all the dephasing times should be larger than 1 ps in order to exclude detrimental effects of dephasing in the proposed experiment.

A. Collisional dephasing
Ultrafast experiments typically use either an effusive beam or supersonic expansion, both of which are hard to model. Instead, one can estimate an upper bound to the collision frequency by using a simple equilibrium gas model at elevated temperatures.
The collisional frequency in the ideal gas at equilibrium is given by [59] 1 where τ coll is the collision period, σ the collisional cross-section, m the mass of the molecule, P pressure, T temperature, and k B Boltzmann's constant. The upper boundary for the τ coll can be estimated by taking the P = 1 atm., T = 300 K, being the distance between Cl and O in COFCl (the largest interatomic distance), and R VdW are the van der Waals radii of the respective atoms [60]. Such estimation gives τ coll ≈ 400 ps, which is much greater than the desired limit of 1 ps.

B. Rotational dephasing
To estimate the timescale of rotational dephasing, we assume rotations and vibrations to be decoupled due to their different characteristic timescales and all vibrational-rotational transitions having the same transition matrix elements. The oscillation of the chiral observable O (Eq. 16) at long times after the pump pulse (t ≫ τ , when (1 + erf(t/τ )) ≈ 2) can be expressed as where I is the intensity of the signal, ω 01 the vibrational transition frequency in the electronic ground state, and A amplitude of the observable's (O) oscillation. However, the vibrational coherence, induced by the laser pulse would include multiple rotational states. In the rigid rotor and harmonic oscillator (RRHO) approximation, the induced rovibrational transition frequency will be given as ω = ω 01 + δω, where δω denotes the difference in the rotational frequencies of the pair of the coherent states. We approximate the rotational states of the asymmetric top by those of a spherical top: Each state |JKM ⟩ with total momentum J = 0, 1, 2, . . ., and projections of the total momentum on the molecular and lab frames K, M = −J, −J + 1, . . . , 0, . . . , J − 1, J has an energy of hBJ(J + 1), where B = ℏ 4πI is the rotational constant, dependent on the moment of inertia I. The rotational energy levels for each J are (2J + 1) 2 times degenerate. We take the levels to be occupied according to Boltzmann distribution and assume that the classical regime is applicable, i.e., thermal fluctuations k B T are more prominent than the difference between nearest quantum levels. With these assumptions, the predicted signal for the molecule in the specific initial and final rotational states will be given by Vibrational-rotational spectra of nonlinear molecules in the electric dipole approximation consist of three branches of transitions with different frequency shifts due to rotational state change: P-branch with ∆J = −1 (|J⟩ → |J − 1⟩): δω P = −4πBJ , Q-branch with ∆J = 0 (|J⟩ → |J⟩): δω Q = 0 , R-branch with ∆J = +1 (|J⟩ → |J + 1⟩): δω R = 4πB(J + 1) .
For the Raman transition, the P-and R-branches have to be replaced by the O-branch (|J⟩ → |J − 2⟩) and S-branch (|J⟩ → |J + 2⟩) but in the classical limit (∆E J→J ′ ≈ J) this will not change the final result.
The final signal will be a combination of interfering signals from each of the three branches, In this approximation, the signal from the Q-branch does not depend on J, whilst the frequencies of the P-and R-branches depend linearly on J. To estimate I P + I R , we take the classical limit, replacing the sum +∞ J=0 by an integral +∞ 0 dJ and assuming J to be large enough such that J(J + 1) ≈ J 2 , (2J + 1) 2 ≈ 4J 2 and J + 1 ≈ J. This gives All the three integrals needed (Z, I ± ) are easy to calculate: where for I ± we have used the substitution J → j ± = J ∓ ikBT t ℏ . When combining all the terms, we obtain The maximal value of Ξ(t) is one (for t = 0)), and the t = +∞ limit is Ξ(+∞) = 1/3, which corresponds to a cancelation of the signals between P and R branches with only the Q branch left. A plot of Ξ(t) for the different rotational As an estimate of the rotational decoherence times, we can use the time of the half-decay τ rot,1/2 for the Ξ(t), defined through the equation Ξ(τ rot,1/2 ) = 2/3. The Figure S7 shows the dependence of the τ rot,1/2 for COF 35 Cl molecule at the different rotational temperatures. A signal lasting longer than 1 ps is expected for rotational temperatures below 70 K. Such temperatures are easily achieved in molecular beam experiments. In the proposed experiment, they are needed also to ensure negligible thermal population in the excited vibrational level of the electronic ground state. We therefore conclude that rotational dephasing will not be relevant.

C. Vibrational excitation intramolecular vibrational energy redistribution lifetime
Coupling of the degree of interest, the OOP vibration, to other vibrational modes will lead to intramolecular vibrational energy redistribution (IVR) and decay of the desired signal. To estimate the rate with which excitation in the OOP vibration decays into other vibrations due to anharmonic couplings, we have performed ab initio molecular dynamics (AIMD) simulations for COFCl. We have taken the ⟨0|ξ|1⟩ average structure obtained at the B3LYP/cc-pVTZ level of theory, cf. Table S1, as the initial structure for the simulations. Then, 28 AIMD trajectories with velocities generated from a Maxwell-Boltzmann distribution at 300 K were calculated for a duration of 0.5 ps with a time step of 0.1 fs, such that the trajectories covered approximately ten periods of OOP vibration in COFCl. The gradients were obtained at the B3LYP-D3/def2-SVP level of theory. From each trajectory, the evolution of the carbon displacement from the ab-plane, was extracted. The results are shown in Figure S8. The average time dependence of this coordinate was fitted to a function describing an harmonic oscillator with friction, with A ξ the vibrational amplitude, ϕ the phase and τ IVR an estimate of the IVR relaxation time, also shown in Figure S8. The latter was found to be τ IVR = 3.6 ± 0.2 ps. This suggests that IVR in COFCl is about two orders of   Table S2 The ground and excited potential energy surfaces of the neutral COFCl (Vg and Ve) and ground state potential of the COFCl cation (Vi) along the out-of-plane vibrational coordinate ξ (see also figure S9) at the B3LYP/cc-pVTZ and TDA-B3LYP/cc-pVTZ levels of theory. The approximation is made using polynomial given in Equation 17. Coefficients Vn are given in cm −1 .
n Vg,n Ve,n Table S3 The ground state dipole moment dependence of the COFCl along the out-of-plane vibrational coordinate ξ at the B3LYP/cc-pVTZ level of theory. The approximation is made using polynomial given in Equation 17. Coefficients